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SIAM J. SCI. COMPUT.
Vol. 38, No. 4, pp. B593–B618
c 2016 Society for Industrial and Applied Mathematics
ACTIVE-SET REDUCED-SPACE METHODS WITH NONLINEAR
ELIMINATION FOR TWO-PHASE FLOW PROBLEMS IN
POROUS MEDIA∗
HAIJIAN YANG† , CHAO YANG‡ , AND SHUYU SUN§
Abstract. Fully implicit methods are drawing more attention in scientific and engineering
applications due to the allowance of large time steps in extreme-scale simulations. When using a
fully implicit method to solve two-phase flow problems in porous media, one major challenge is the
solution of the resultant nonlinear system at each time step. To solve such nonlinear systems, traditional nonlinear iterative methods, such as the class of the Newton methods, often fail to achieve
the desired convergent rate due to the high nonlinearity of the system and/or the violation of the
boundedness requirement of the saturation. In the paper, we reformulate the two-phase model as
a variational inequality that naturally ensures the physical feasibility of the saturation variable.
The variational inequality is then solved by an active-set reduced-space method with a nonlinear
elimination preconditioner to remove the high nonlinear components that often causes the failure of
the nonlinear iteration for convergence. To validate the effectiveness of the proposed method, we
compare it with the classical implicit pressure-explicit saturation method for two-phase flow problems with strong heterogeneity. The numerical results show that our nonlinear solver overcomes the
often severe limits on the time step associated with existing methods, results in superior convergence performance, and achieves reduction in the total computing time by more than one order of
magnitude.
Key words. two-phase flow, variational inequality, active-set reduced-space methods, nonlinear
preconditioners, nonlinear elimination, parallel computing
AMS subject classifications. 76S05, 68W10
DOI. 10.1137/15M1041882
1. Introduction. Modeling and simulation of two-phase flow in porous media
have been a major effort in reservoir engineering [37]. In addition to oil reservoir
management, understanding and modeling of two-phase flow are crucial also to many
environment issues. For example, one of the most attractive and practical solutions
to reducing the CO2 emission problem is to inject and store CO2 in subsurface geological formations [43], such as depleted reservoirs and deep saline aquifers. The large
capacity of subsurface storage provides several advantages over other possible alternatives of carbon sequestration. For subsurface carbon sequestration, there are at
least four major mechanisms to trap the injected CO2 for long-term storage, namely,
structural (stratigraphic) trapping, residual fluid trapping, solubility trapping, and
mineral trapping. Some of these mechanisms may coexist and interplay simultaneously during the long span of storage process. A deep understanding of these trapping
mechanisms and exploring possibly new trapping mechanisms require accurate mod∗ Submitted to the journal’s Computational Methods in Science and Engineering section September 29, 2015; accepted for publication (in revised form) May 17, 2016; published electronically July
26, 2016. The research was supported in part by NSF China grants 11571100, 91530323 and 91530103.
http://www.siam.org/journals/sisc/38-4/M104188.html
† College of Mathematics and Econometrics, Hunan University, Changsha, Hunan, 410082, China
([email protected]).
‡ Corresponding author. Institute of Software, Chinese Academy of Sciences, Beijing 100190,
China, and State Key Laboratory of Computer Science, Chinese Academy of Sciences, Beijing 100190,
China ([email protected]).
§ Division of Physical Sciences and Engineering (PSE), King Abdullah University of Science and
Technology (KAUST), Thuwal 23955-6900, Saudi Arabia ([email protected]). This author
was supported by KAUST through the grant BAS/1/1351-01-01.
B593
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B594
HAIJIAN YANG, CHAO YANG, AND SHUYU SUN
eling and simulation of two-phase and multiphase flow in subsurface formations. This
paper is restricted to the discussion of two-phase flow, but the applied methodology
is applicable to full compositional multiphase flow.
One basic requirement for accurate modeling and simulation of multiphase flow is
to have the predicted saturation sit within a physically meaningful range, i.e., between
zero and one. The saturation can be spatially and temporally varied within this
range, but most algorithms used in practice do not guarantee the computed saturation
will stay within this range. This is especially so when the applied computational
method manifests numerical oscillation and nonphysical undershoot or overshoot [33].
A commonly used fix to this problem is to simply apply a cut-off operator to the
computed saturation at each time step, i.e., to set the saturation to be zero whenever
it becomes negative, and to set it to one whenever it becomes larger than one. This
treatment, however, has a serious unpleasant consequence, which is the loss of mass
conservation. Global and local mass conservation are desired properties for reservoir
flow and other subsurface flow simulation. The majority of algorithms used in reservoir
simulation have the property of local mass conservation, which of course implies also
global conservation. Unfortunately, the applied cut-off operators not only destroy the
local mass conservation but also damage the global mass conservation, which seriously
ruins the numerical accuracy and physical interpretability of the simulation results.
It is safe to state that one major challenge in two-phase flow simulation is the
nonlinearity of the process. Two main nonlinearities of two-phase flow are capillarity
and relative permeability. Capillary pressure is one of main driving forces in fluid
(oil, gas, and water) flow and transport in subsurface [37]. The capillarity effect can
be the leading mechanism in oil recovery from fractured oil reservoirs. Capillarity is
caused by surface tension between immiscible (or partially miscible) fluids. Relative
permeability is a dimensionless measure of the effective permeability of a fluid phase,
and it also plays an important role in enhanced oil recovery. In commonly used
multiphase flow models, both the capillary pressure and relative permeability are
nonlinear functions of saturation. They can be defined by interpolating experimental
data or by using a closed-form formula, and they usually require the saturation to
stay within a physically meaningful range, because otherwise the capillary pressure
and relative permeability become undefined. As a result, it is important to ensure the
saturation within a physically meaningful range, even from the viewpoint of treating
capillarity and relative permeability.
A popular approach for solving two-phase flow problems is the implicit pressureexplicit saturation (IMPES) method [11, 12, 19, 59], in which the nonlinear problem
is decoupled into two subproblems, namely, a linear equation for the potential component that needs to be solved implicitly and a nonlinear equation for the saturation
variable that can be explicitly updated. However, the IMPES method often requires
use of a small time step size to maintain a physical meaningful solution because of the
stability restriction [13]. The situation becomes worse at the extreme scale due to the
dependency between the time step size and the spatial resolution. Moreover, the semiimplicit nature of the algorithm may introduce a large splitting error due to the decoupling of the original problem [53]. Fully implicit methods [14, 36, 42, 51, 53, 54, 57],
on the other hand, can relax the stability requirement on the time step size and therefore provide a consistent and robust way to solve the nonlinear problem in long-term
and extreme-scale simulations. In particular, for problems with various time scales,
fully implicit methods with a suitable time adaptivity strategy [44, 58] can lead to
substantial reduction the total computing time, which greatly expands the scope of
application of the fully implicit algorithm.
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NONLINEAR ELIMINATION FOR TWO-PHASE FLOW
B595
When a fully implicit method is applied, one needs to solve a large-scale nonlinear
algebraic system at each time step. Efforts have been made in employing the classical
Newton method and its many variants to solve the two-phase flow problem [14, 36, 42].
However, numerical challenges arising from the boundedness of the solution still exist
and may cause the Newton methods fail to converge. In this work, we introduce a
variational inequality formulation of the saturation equilibrium with a box inequality
constraint, which naturally satisfies the boundedness requirement. The two-phase
variational inequality is then formulated as a nonlinear complementarity problem
[21] which is solved by an active-set reduced-space method [3, 4, 22]. The active-set
reduced-space method consists of two major steps: (a) in the first phase, an index
set with respect to the computational domain is decomposed into active and inactive
parts, based on a criterion specifying a certain active set method; and (b) in the
second phase, a reduced linear system associated with the inactive set is solved. We
would like to point out that the class of active-set reduced-space method has proven
to be very efficient in a variety of applications [23, 28, 34, 35, 56], but very little work
has been done in the two-phase flow problem as far as we know.
For problems with high nonlinearities, preconditioning techniques on the nonlinear level, such as the additive Schwarz preconditioned inexact Newton (ASPIN) methods [9, 25, 39], the nonlinear restricted Schwarz preconditioners [10, 16], the nonlinear
dual-domain decomposition methods [48], the nonlinear balancing domain decomposition by constraints methods [30], the nonlinear elimination methods [26, 27, 29],
and the composite nonlinear algebraic methods [6], have received increasing attention in recent years. In particular, some efforts have been made in applying the
ASPIN method to solve the two-phase flow problems [51, 53]. Although the numerical results show that it is superior to the classical linear additive Schwarz method,
the success of the ASPIN method depends strongly on an effective nonlinear subdomain solver, which limits the robustness of the algorithm, especially for the highly
nonlinear models of two-phase flow problems. In this study, we present a nonlinear
elimination algorithm as a nonlinear preconditioner to accelerate the convergence of
the active-set reduced-space method. In the algorithm, the nonlinear elimination is
carried out in a field-split manner to implicitly eliminate the solution components
with high magnitude that often triggers the divergence of the nonlinear solver. We
show by numerical experiments that the proposed algorithm is robust and efficient for
a number of two-phase flow examples with strongly nonlinear relative permeabilities
and spatially varied capillary pressure functions.
The paper is organized as follows. In section 2, we present a detailed description
of the active-set reduced-space method with nonlinear elimination. In section 3, a
variational inequality based model of two-phase flow problems is introduced and then
the proposed algorithm is applied to solve the resultant nonlinear systems. We carry
out numerical experiments in section 4 to examine the efficiency and effectiveness of
the proposed algorithm. The paper is concluded in section 5.
2. Active-set reduced-space method with nonlinear elimination. The
proposed active-set reduced-space method with nonlinear elimination consists of two
major ingredients: (a) a global update based on an active-set reduced-space method as
an outer nonlinear solve and (b) a subspace correction based on a field-split nonlinear
elimination as a nonlinear preconditioner.
2.1. Active-set reduced-space methods. We first review the class of activeset reduced-space methods (see, e.g., [2, 45]) for the numerical solution of large spare
nonlinear variational inequality system of equations with lower and upper bounds,
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B596
HAIJIAN YANG, CHAO YANG, AND SHUYU SUN
i.e., for a given nonlinear function F : Rn

 Xi = φi
Xi = ψi
(2.1)

Xi ∈ [φi , ψi ]
→ Rn , compute a vector X ∈ Rn , such that
and Fi (X) > 0,
and Fi (X) < 0,
and Fi (X) = 0,
where F = (F1 , . . . , Fn )T , Fi = Fi (X1 , . . . , Xn ), X = (X1 , . . . , Xn )T , and the inequality holds componentwisely by φ = (φ1 , . . . , φn )T and ψ = (ψ1 , . . . , ψn )T , i.e., only one
of these three equations holds at a time. Let S = {1, 2, . . . , n} be an index set with
each index corresponding to an unknown component Xi and a nonlinear residual
component Fi . Two special cases are the system of nonlinear equations, F (X) = 0,
obtained by taking φi = −∞ and ψi = +∞ for each i ∈ S, and the mixed nonlinear
problem, where for some i in the subset of S, φi = −∞ and ψi = +∞. In the study, we
propose a class of active-set reduced-space methods for solving (2.1), which consists
of the following steps: (a) in the first phase, the index set is decomposed into active
and inactive parts, based on a criterion specifying a certain active set method; (b) in
the second phase, a reduced linear system associated with the inactive set is solved,
and then a line search method is applied to improve the efficiency of the iterative
process.
The active and inactive sets used within the reduced-space method are defined as

 Aφ (X) := {i ∈ S | Xi = φi and Fi (X) > 0} ,
Aψ (X) := {i ∈ S | Xi = ψi and Fi (X) < 0} ,

I(X) := S\ (Aφ (X) ∪ Aψ (X)) ,
where Aφ (X) and Aψ (X) denote the active constraints at X with respect to the lower
bound φ and the upper bound ψ, respectively, and I(X) the inactive constraints. In
fact, the active sets denote the variables where the lower and upper bounds are active
and the function value can be ignored, i.e.,
φi if i ∈ Aφ (X),
Xi =
ψi if i ∈ Aψ (X).
The inactive set contains the remainder of the variables associated with a reduced
linear system.
Starting from an initial guess X 0 ∈ Rn , suppose X k is the current approximate
solution; then a new approximate solution X k+1 can be computed through
X k+1 = π X k + λk dk ,
where dk is a search direction, λk is the step length for line search, and the operator π
is a projection to cut off the undershoot or overshoot of the solution from the interval
[φ, ψ], i.e., to set the component of the solution to be the lower bound whenever it
becomes less than φ, and to set it to ψ whenever it becomes larger than the upper
bound. To be more precise, at every iteration of the reduced-space method, the
search direction dk includes three mutually disjoint subvectors dAkφ , dAkψ , and dI k ,
with the first two corresponding to the active sets and the third to the inactive set.
In the computation, we first set dAkφ and dAkψ to 0, and then dI k is calculated by
approximately solving the linear system equation
(2.2)
∇F (X k ) I k ,I k dI k = −FI k (X k ),
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NONLINEAR ELIMINATION FOR TWO-PHASE FLOW
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Algorithm 1. Active-set reduced-space method (RS).
Step 1 Choose an initial guess X 0 ∈ [φ, ψ]
and set k = 0.
Step 2 While kFΘ (X k )k > εr kFΘ (X 0 )k and kFΘ (X k )k > εa do
• Define the active and inactive sets

 Aφ (X) := {i ∈ S | Xi = φi and Fi (X) > 0},
Aψ (X) := {i ∈ S | Xi = ψi and Fi (X) < 0},

I(X) := S\ (Aφ (X) ∪ Aψ (X)) .
• Set dAkφ = 0 and dAkψ = 0, and approximately solve the linear system
∇F (X k )
I k ,I k
dI k = −FI k (X k )
to calculate a direction
dIk .
• Set X k+1 = π X k + λk dk , where λk ∈ (0, 1] is determined to satisfy
FΘ π X k + λk dk k2 6 (1 − αλk )kFΘ (X k ) .
2
• Set k = k + 1.
End While
where ∇F (X k ) I k ,I k is a submatrix of ∇F (X k ) and FI k (X k ) is a subvector of
F (X k ), both based on the same index set I k . The step size λk is chosen to be the
optimal value for residual reduction with the constraint of
(2.3)
kFΘ π X k + λk dk k2 6 (1 − αλk )kFΘ (X k )k2 ,
where α is employed to ensure that the reduction of kF (X k )k2 is sufficient, and FΘ (X)
is a restriction operator of F (X) defined componentwise as
Fi (X)
if φi < Xi < ψi ,
[FΘ (X)]i =
min{Fi (X), 0} others.
Now the complete active-set reduced-space method (RS) algorithm is described in
Algorithm 1.
In the RS method, we obtain a search direction dk by solving inexactly the reduced
linear system (2.2), then compute the next approximate solution along this search
direction. In (2.3), the damping scalar λk is used to determine the step length that
one should go in the selected search direction, and the parameter α is used to ensure
that the reduction of the residual function kFΘ (X)k2 is sufficient. In the ideal case,
λk = 1 is sufficient for RS, which means that a full step is taken. A near quadratic
convergence may be observed, when the nonlinearities in the system are well-balanced.
However, when the nonlinearity of the system is severely imbalanced, the size of
λk could be very small and the convergence becomes much slower. Observed from
many numerical experiments, the slow convergence or sometimes divergence is often
determined by the variables of equations in the system with the highest nonlinearities
[9, 10, 25]. In our proposed algorithm, a nonlinear elimination step is applied as a
subproblem solver inside the global RS iteration to smooth out the “high nonlinearity.”
Through subspace nonlinear solves, the high nonlinearities are removed, and the fast
convergence can then be restored when the active-set reduced-space method is applied
after the preconditioning.
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B598
HAIJIAN YANG, CHAO YANG, AND SHUYU SUN
2.2. Nonlinear elimination. To define the nonlinear elimination (NE) preconditioner that is based on a field partition, the index set S is decomposed into two
parts Sb and Sg corresponding to different physical or algorithmic components, i.e.,
S = Sb ∪ Sg . For notational convenience, we denote the vector X = (XSb , XSg ), and
the nonlinear system F (X) is rewritten as
FSb (XSb , XSg )
F (X) = F (XSb , XSg ) =
= 0.
FSg (XSb , XSg )
For this partition, we define the subspace corresponding to the set Sb , Vb ⊂ Rn
as
(2.4)
Vb = {v | v = (v1 , . . . , vn )T ∈ Rn , vi = 0, if i 6∈ Sb },
and the corresponding restriction operators Rb , which transfer a vector from Rn to
Vb . Similarly, we define the subspace
(2.5)
Vg = {v | v = (v1 , . . . , vn )T ∈ Rn , vi = 0, if i 6∈ Sg },
and the corresponding operator Rg , which restrict the vector from Rn to Vg . Hence,
using the restriction operator Rb , for any given X ∈ Rn , we have
FSb (X) = Rb (F (X)) .
Moreover, we define Tb (X) : Rn → Vb as the solution of the following subspace
nonlinear system:
(2.6)
FSb (Rg (X) + Tb (X)) = 0.
Using the subspaces mapping function, we introduce a new global function,
Y = G(X) = Rg (X) + Tb (X).
2.3. General framework of RS–NE. The proposed nonlinear elimination preconditioned active-set reduced-space method can be seen as a class of nonlinear rightpreconditioning methods, i.e., the nonlinear elimination method is proposed for the
construction of the inner nonlinear preconditioner and the classical active-set reducedspace method is used for the outer nonlinear solver. A high-level description of the
basic algorithm for a general problem is presented in Algorithm 2.
The basic idea of RS–NE is to approximately eliminate the high nonlinearities
before applying the global nonlinear iteration. Skipping the subspace correction phase,
the proposed method is reduced to the classical RS method. Moreover, the RS method
degrades into the classical inexact Newton method with backtracking [15, 17] for
solving (2.1) when φ = −∞ and ψ = +∞. Hence, for the mixed nonlinear problem,
where for some i in the subset of S, φi = −∞ and ψi = +∞, the proposed method
can be considered as a class of coupled methods, i.e., the inexact Newton method with
backtracking is used to solve the part of nonlinear system without restrictions, while
the RS method is employed to solve the part of nonlinear system with restrictions. In
Algorithm 2, a parameter switch , which is problem-dependent and requires the user to
set, is employed as a switch to determine the timing of turning on or off the subspace
correction phase. In addition, if the line search fails in the process of global nonlinear
iterations, then the RS–NE method reverts to the subspace correction phase.
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NONLINEAR ELIMINATION FOR TWO-PHASE FLOW
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Algorithm 2. Active-set reduced-space method with nonlinear elimination (RS–NE).
Step 1 Choose an initial guess X 0 ∈ [φ, ψ]
and set k = 0.
Step 2 While kFΘ (X k )k > εr kFΘ (X 0 )k and kFΘ (X k )k > εa do
Global update phase:
• Define the active and inactive sets

 Aφ (X) := {i ∈ S | Xi = φi and Fi (X) > 0},
Aψ (X) := {i ∈ S | Xi = ψi and Fi (X) < 0},

I(X) := S\ (Aφ (X) ∪ Aψ (X)) .
• Set dAkφ = 0 and dAkψ = 0, and approximately solve the linear system
∇F (X k ) I k ,I k dI k = −FI k (X k )
to calculate a direction dI k .
• Set X k+1 = π X k + λk dk , where λk ∈ (0, 1] is determined to satisfy
kFΘ π X k + λk dk k2 6 (1 − αλk )kFΘ (X k )k2 .
• Set k = k + 1.
Subspace correction phase:
If (line search fails) or (switch == 1), then
• Given X k = (Xbk , Xgk ), find Tb (X k ) by solving the subspace problem
FSb (Rg (X k ) + Tb (X k )) = 0.
• Compute Y k = G(X k ) = Rg (X k ) + Tb (X k ), and update X k = Y k .
End if
End While
It is worth mentioning that the proposed nonlinear elimination method is based
on the nonlinear function F (X), rather than the variational inequality (2.1), to build
the subspace systems (2.4) and (2.5). Moreover, the subspace problem (2.6), which
is a reduced-space nonlinear system, can be solved using any classical nonlinear iterative method, such as the inexact Newton method with backtracking and its many
variations [7, 31, 46, 47], and other composite solvers [6]. A key element of the NE
step is the choice of Sb and Sg to build the subspaces Vg and Vb . Some physics-based
or field-based strategy can be used to determine the subset indices, and the partition
of Sb and Sg for the nonlinear elimination may change with k. It differs from case to
case. We also remark that the use of the nonlinear elimination preconditioner in this
paper is different from the work in [26, 27] for the one- and two-dimensional scalar
full potential equations. In [26, 27], the to-be-eliminated component is determined in
advance based on certain knowledge of the problem, and thus the NE preconditioner
is applied starting from the beginning until the intermediate solution is close to the
desired solution. In our study, a class of nonlinear elimination methods as a right preconditioner is used for the general multicomponent systems, and the to-be-eliminated
component is implicitly removed, based on the the feedback from the intermediate
global nonlinear iterations. Hence, the NE preconditioner is activated only when the
line search fails or the control parameter switch is triggered in case that the global
update is difficult to converge.
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B600
HAIJIAN YANG, CHAO YANG, AND SHUYU SUN
3. RS–NE for two-phase flow in porous media. In this section, we introduce the continuous physical model together with the discretization employed for
two-phase flow problems in porous media, and then present the implementation details
of the RS–NE method for the resultant nonlinear systems.
3.1. Model problem. The flow consisting of two incompressible and immiscible
fluids in porous media can be described by

∂S

φ α + ∇ · vα = qα ,
∂t
(3.1)
α = w, n,
k k

vα = − rα (∇pα + ρα g∇z) ,
µα
where g is the gravity acceleration constant, z is the depth, and φ and k are the
porosity and the absolute permeability of the media. Here the wetting phase of the
flow is denoted by subscript w and the nonwetting phase n. Each phase α has its
own saturation Sα , velocity vα , pressure pα , density ρα , viscosity µα , and relative
permeability krα . Later, we will show that the primary variables to be solved in (3.1)
are Sw and pw . In this paper, the relative permeabilities are given by
krw = Seβ ,
krn = (1 − Se )β ,
where β is a positive integer number and Se is the normalized saturation. A commonly
employed formulation of the normalized saturation is
Se =
Sw − Srw
,
1 − Srw − Srn
where Srw and Srn are the residual saturations for the wetting and nonwetting phases,
respectively.
In (3.1), the first equation is determined by the mass conservation law, and the
second one is the velocity equation following the Darcy’s law. For the two-phase flow,
the saturations of the phases are constrained by
Sw + Sn = 1,
and the pressures of the two phases are related through the capillary pressure function
pc (Sw ) = pn − pw .
Throughout the study, we use the following nonlinear capillary pressure function [24]:
Bc
pc (Sw ) ≡ − √ log Se ,
k
where Bc > 0 is related to the media property.
The heterogeneity of the capillary pressure in permeable media may result in
discontinuity in the saturation, which in turn complicates the numerical simulation.
For example, in the classical fractional flow formulation, contrast in capillary pressure
may lead to discontinuous global pressure that makes the formulation inconsistent. In
this paper, we follow the formulation in [24, 41]. We first introduce the flow potential
Φα for phase α:
Φα = pα + ρα gz.
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NONLINEAR ELIMINATION FOR TWO-PHASE FLOW
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The capillary potential is then defined as
(3.2)
Φc ≡ Φn − Φw = pc (Sw ) + (ρn − ρw )gz.
Further, we denote the mobility of phase α as λα = krα /µα . The total velocity can
be rewritten into
v ≡ vw + vn = va + vc
with
(3.3)
va = −λa k∇Φw ,
vc = −λn k∇Φc .
Here the velocity va is associated with a mobility λa = λw + λn that is smoother compared to either λw or λn . The wetting-phase velocity can be recovered as vw = fw va
with fw = λw /λa . Then, the potential equation of the two-phase flow is reformulated as
(3.4)
∇ · (va + vc ) = qw + qn ,
and the wetting-phase saturation equation becomes
(3.5)
φ
∂Sw
+ ∇ · (fw va ) = qw ,
∂t
which holds on the set 0 6 Sw 6 1.
Suppose the boundary of the computational domain Ω is composed of two parts
∂Ω = ΓD + ΓN with ΓD ∩ ΓN = ∅. The boundary conditions associated to (3.4) and
(3.5) are

on ΓD ,
 pw = pD
N
v·n=q
on ΓN ,

N
Sw = S
on ∂Ω.
And the initial condition for the saturation equation is
0
(3.6)
Sw t=0 = Sw
in Ω.
When using a fully implicit method, (3.4) and (3.5) are coupled and a nonlinear
system is required to be solved at each time step. However, if we directly apply the
Newton method or its many variations to solve such nonlinear systems, we face the
numerical challenges arising from physically feasible saturation fractions between 0
and 1. Hence, we take the restriction of the saturation into account and introduce a
variational inequality problem of the wetting-phase saturation equation (3.5). Considering a scaled L2 -inner product and the obstacle potential 0 6 Sw 6 1, we obtain
the following variational inequality for (3.5); see [4, 5, 22] for more details.
0
0
Problem 1. Given Sw (·, 0) = Sw
∈ H 1 (Ω) with 0 6 Sw
6 1, find Sw ∈ H 1 (ΩT )
such that 0 6 Sw 6 1 a.e. in ΩT = Ω × [0, T ] and
(3.7)
D ∂S
E D
E D
E
w
φ
, X − Sw + ∇ · (fw va ) , X − Sw − qw , X − Sw > 0,
∂t
which has to hold for almost all t and all X ∈ H 1 (Ω) with 0 6 X 6 1. Here h·, ·i
denotes the L2 -inner product.
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It is well known that the above problem is equivalent to the following problem:
(3.8)

∂Sw


+ ∇ · (fw va ) − qw > 0
Sw = 0, φ


∂t





∂Sw
Sw = 1, φ
+ ∇ · (fw va ) − qw < 0

∂t







 Sw ∈ [0, 1] , φ ∂Sw + ∇ · (fw va ) − qw = 0
∂t
a.e. in
ΩT ,
a.e. in
ΩT ,
a.e. in
ΩT ,
where all equalities and inequalities have to hold almost everywhere. Equation (3.8) is
known as a box constrained variational inequality or mixed complementarity problem.
For more details, we recommend [18, 20, 21, 45, 49] and references therein.
For the purpose of comparison, an IMPES method is also implemented. The
semi-implicit scheme comprises two split steps: one implicitly solves the potential
equation (3.4) to obtain a new potential of the wetting phase Φw and thus va from
(3.3), and the other explicitly updates the wetting-phase saturation (3.5).
3.2. Discretization. In this study, a cell-centered finite difference (CCFD) is
applied to spatially discretize the optimization problem for the spatial terms; the detailed process of CCFD can be found [32, 40, 42]. The employed CCFD discretization
for the pressure equation in rectangular meshes can also be viewed as the mixed finite element method with Raviart–Thomas basis functions of lowest order equipped
with the trapezoidal quadrature rule. We remark here that the proposed nonlinearly
preconditioned solutions algorithms can be generalized to the case of other spatial
discretizations.
Let Ω be the computational domain covered with Nx × Ny mesh cells. Then each
mesh point pi,j = (xi , yj ) is centered at the position xi = i × hx and yj = j × hy with
i = 1, . . . , Nx , j = 1, . . . , Ny , hx = 1/Nx , and hy = 1/Ny . For a given time-stepping
sequence 0 = t(0) < t(1) < t(2) < · · · , define the time step size ∆t(l) = t(l+1) − t(l) and
use superscript (l) to denote the discretized evaluation at time level t = t(l) . Then,
the discretization of the saturation equation in (3.5) is
(3.9)
(l)
(l+1)
− Sw
Sw
∗
(l+1) (l+1)
(l+1) (l+1)
(l+1)
FSw Sw
+
∇
·
f
v
, pw
− qw
= 0,
=φ
h
w
a
∆t(l)
where ∇∗h · represent CCFD discretization of the gradient operator by using the up(l+1)
wind scheme according to the velocity va
. Hence, the discretization of variational
inequality (3.8) is given as
(3.10)

(l+1)

Sw
=0







(l+1)
Sw
=1







(l+1)
 Sw
∈ [0, 1]
(l+1) (l+1)
and FSw Sw , pw
> 0,
(l+1) (l+1)
and FSw Sw , pw
< 0,
(l+1) (l+1)
and FSw Sw , pw
= 0.
Similarly, we employ a CCFD scheme for (3.4); then the discretization of the
potential equation is
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NONLINEAR ELIMINATION FOR TWO-PHASE FLOW
B603
(3.11)
(l+1) (l+1)
(l+1)
= −∇h · λ(l+1)
Fpw Sw
, pw
∇h Φ(l+1)
− ∇h · λ(l+1)
∇h Φ(l+1)
− qw
− qn(l+1)
a
w
n
c
= 0,
where ∇h · and ∇h represent CCFD discretizations of the divergence and gradient
operators. We remark here that the CCFD scheme conserves the mass both globally
and locally because of the consistent discretization of the local differential operator
across all mesh cell interfaces.
3.3. The application of RS–NE to two-phase flow. In this subsection, we
introduce the implementation details of the RS–NE method for two-phase flow. In the
implementation, we use the fully coupled ordering to build up the nonlinear system
(2.1), by which we mean that different physical variables defined at the same grid
point are always put together throughout the calculations. At each grid point, we
arrange the unknowns in the order of (pw )ij and (Sw )ij , and then all grid points
numbered in the natural ordering. That is, the unknowns are ordered in the order of
(3.12) X = (pw )11 , (Sw )11 , . . . , (pw )ij , (Sw )ij , . . . , (pw )Nx Ny , (Sw )Nx Ny ∈ Rn ,
where n = 2Nx Ny . The corresponding functions are ordered in the order of
(3.13)
F (X) = (Fpw )11 , (FSw )11 , . . . , (Fpw )ij , (FSw )ij , . . . , (Fpw )Nx Ny , (FSw )Nx Ny ∈ Rn .
The lower bound in (2.1) is given as
φ = (−∞, 0, . . . , −∞, 0, . . . , −∞, 0) ∈ Rn ,
and the upper bound is
ψ = (+∞, 1, . . . , +∞, 1, . . . , +∞, 1) ∈ Rn .
In the NE step, we use a field-based strategy to determine Sb and Sg and propose
the inexact Newton method with backtracking for solving the subspace problem (2.6).
In the following, we take the wetting-phase saturation approach (i.e., removing the
Sw -component) as an example to build the NE step. In the approach, we consider all
the wetting-phase saturation Sw as bad components. Following the order in (3.12),
the index set Sb with respect to the component Sw is all the even numbers of the set
S, and Sg = S\Sb . Also, we can define the corresponding subspaces Vb and Vg in
(2.4) and (2.5), respectively. Then,
Xi
if i ∈ Sg ,
Yi =
(Tb (X))i if i ∈ Sb ,
where Tb is the solution of subspace nonlinear system (2.6) with the inexact Newton
method with backtracking. To active the NE step, we need to set the parameter
switch = 1 when necessary. The strategy we use here is as follows. We first take
several (controlled by a threshold Nswitch , unless the line search fails) global nonlinear
iterations using the RS method, then solve the subspace problem by the inexact
Newton method with backtracking, and this process continues until convergence.
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B604
HAIJIAN YANG, CHAO YANG, AND SHUYU SUN
A major advantage of fully implicit schemes is that the time step size 4t is
no longer constrained by the CFL condition, which is often required by explicit or
semi-implicit methods. Meanwhile, for the simulation of two-phase flow, the evolution
of the system usually admits various time scales and the calculation often lasts for a
long time. Hence, we can use an adaptive time step control to enhance the temporal
accuracy and reduce the total computing time. In the study, analogous to the switched
evolution/relaxation strategy in [44], we start with a relatively small time step size
4t(0) and update its value via 4t(l+1) = β (l) 4t(l) with β (l) being the scaling factor
of the adjacent time step size
η !!
kF (X (l) )k2
1
(l)
, min ρ,
,
(3.14)
β = max
ρ
kF (X (l+1) )k2
where kF X (l+1) k2 is the Euclidean norm of F X (l+1) . In the formulation, ρ ∈
(0, +∞) is a safeguard to avoid excessive change of the time step size between any
two immediate time steps, and η ∈ (0, 1] is used to control the adjustment of the time
step size.
4. Numerical experiments. In this section, we report some results of numerical experiments to examine the performance of both traditional and the proposed
algorithms listed as follows:
• IMPES denotes the implicit pressure-explicit saturation method.
• INB denotes the inexact Newton method with backtracking.
• RS denotes the classical active-set reduced-space method.
• RS–NE denotes the active-set reduced-space method with nonlinear elimination.
• Adaptive RS–NE denotes the nonlinear elimination preconditioned active-set
reduced-space method with adaptive time stepping.
All these algorithms are implemented based on the Portable, Extensible Toolkits for
Scientific computation (PETSc) [1] library. We test the algorithms on a Dell PowerEdge C6100 supercomputer that contains two hex-core 2.8-GHz Intel Westmere
processors and 24 GB local memory in each node and equipped with a nonblocking
QDR Infiniband high performance network. It is worth mentioning that all algorithms except the IMPES method are employed in conjunction with the fully implicit
method. There are several nonlinear and linear iterative procedures in the proposed
algorithms, and each requires a proper stopping condition. In the RS method, an
absolute (relative) tolerance of 10−3 (10−8 ) is utilized for the global nonlinear iteration, and the absolute (relative) tolerance is set to 10−10 (10−6 ) for the subspace
nonlinear iteration. In nonlinear iterations, we use the standard cubic backtracking
algorithm [15, 17] with α = 10−4 to pick the step length λ. In the IMPES method,
the time step subdivision for saturation is 5. All linear systems are solved by the
Schwarz preconditioned GMRES method [8, 50, 52, 55], with relative and absolute
tolerances of 10−3 and 10−6 , respectively. The subdomain problems in the Schwarz
preconditioner are solved with the sparse LU factorization. Moreover, in the adaptive
RS–NE method we use ρ = 2.5 and η = 0.75 to control the time step size 4t, if not
explicitly stated.
In the study, we use five test cases for two-phase fluid flows in a horizontal layer
consisting of regular heterogeneous permeability [24, 32, 53]. In all cases, with the
porous media being horizontal, the effect of gravity is neglected, and the void of the
medium is initially fully saturated with oil, i.e., the initial conditions are pw = 0 and
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NONLINEAR ELIMINATION FOR TWO-PHASE FLOW
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Table 1
Parameters for all test cases.
Parameters
Computational domain (m)
φ
k (md)
µw (cp)
µn (cp)
β
Bc (bar md1/2 )
Srw
Srn
Injection rate (PV/year)
Case-1
300 × 150
0.2
1 or 100
1
0.45
2
70
0
0
0.15
Case-2
300 × 150
0.2
1, 50 or 100
1
0.25
2
30
0
0
0.11
Case-3
300 × 150
0.2
1 or 100
1
0.5
2
100
0
0
0.2
Case-4
500 × 270
0.2
1 or 100
1
0.45
2
50
0
0
0.11
Case-5
100 × 100
0.2
0.03-400
1
0.2
2
25
0
0
0.1
Sw = Sm , where Sm = 10−4 is the minimum of saturation. We assume the top and
bottom boundaries of the reservoir to be impermeable, i.e., the normal component
of the Darcy velocity on these boundaries vanishes. Then the system is flooded by
w
water from left to right, i.e., we set ∂p
∂n = 0, Sw = 1 − Sm at the left boundary
∂Sw
and set pw = 0, ∂n = 0 at the right boundary. There is no injection/extraction
to the interior of the domain. Table 1 summarizes the main features of each test
case, including the domain size, permeability, and porosity data. The purposes of
the tests include (a) the verification of the numerical accuracy of the fully implicit
method; (b) a comparison of the performance of the IMPES method and the RS–NE
method; (c) a comparison of different fully implicit solvers, including the classical
INB and RS methods and the RS–NE method with adaptive time stepping; (d) a
comparison of different parameters for RS–NE with adaptive time stepping; (e) the
parallel performance of the fully implicit method.
4.1. Numerical validation. We first validate the discretization scheme and
the fully implicit solver by running Case-1. As shown in Figure 1, the computational
domain for Case-1 is 300 meters long by 150 meters wide and includes two kinds of
permeabilities (1 md and 100 md) in subdomains, respectively, i.e.,
(
100 md 70 6 x 6 150 and 50 6 y 6 100,
k=
1 md
otherwise.
The saturation is calculated until 0.5 pore volume injection (PVI), and the wetting
phase is injected with the flow rate 0.15 PV/year from the left boundary, i.e., the
simulation time is ended at t = 0.5/0.15 year. In this test, we use a fixed time step
size 4t = 0.1 year (i.e., 36.5 days) for the RS–NE method and use 4t = 5 × 10−4
year (i.e., 0.1825 day) for the IMPES method. Numerical results of both the IMPES
method and the fully implicit method on a 100×50 mesh are shown in Figure 1, which
verifies the consistency between the semi-implicit and the fully implicit methods. We
remark that the IPMES method cannot obtain the smooth results when the time step
size 4t is larger than 6 × 10−4 year, which is consistent with [32].
We further run Case-2 to verify the robustness of the fully implicit method. Compared to the previous test case with two different permeabilities, in Case-2 there are
three subdomains with the permeabilities of 1 md, 50 md, and 100 md, as shown
in the top panel of Figure 2. We run the test on a 100 × 50 mesh using the fully
implicit and semi-implicit methods. In the simulation, we again use a fixed time step
size 4t = 0.1 year for the fully implicit method and use 4t = 5 × 10−4 year for the
IMPES method. The wetting phase is injected with the flow rate 0.11 PV/year from
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B606
HAIJIAN YANG, CHAO YANG, AND SHUYU SUN
Fig. 1. Case-1: Heterogeneous permeabilities and wetting-phase saturation profiles at 0.5 PVI.
The mesh is 100 × 50. The top-left panel shows the distribution of the heterogeneous permeabilities;
the top-right panel shows the contour plot of the wetting-phase saturation by using RS–NE with
4t = 0.1 year; the bottom panel shows the contour plot of the wetting-phase saturation by using
IMPES with 4t = 5 × 10−4 year.
Fig. 2. Case-2: Heterogeneous permeabilities and wetting-phase saturation profiles at 0.6 PVI.
The mesh is 100 × 50. The top-left panel shows the distribution of the heterogeneous permeabilities;
the top-right panel shows the contour plot of the wetting-phase saturation by using RS–NE with
4t = 0.1 year; the bottom panel shows the contour plot of the wetting-phase saturation by using
IMPES with 4t = 5 × 10−4 year.
the left boundary. The wetting-phase saturation profiles as 0.5 PVI computed by the
two methods are presented in the bottom panel of Figure 2. The results illustrate
that the new method can yet compute a reasonable solution with a much larger time
step size, when compared with the IMPES approach.
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NONLINEAR ELIMINATION FOR TWO-PHASE FLOW
B607
Fig. 3. Case-3: Heterogeneous permeabilities and wetting-phase saturation profiles by using RS–
NE. The mesh is 100 × 50. The top-left panel is for the heterogeneous permeabilities; the top-right
panel is for wetting-phase saturation profiles at 0.2 PVI; the bottom-left panel is for wetting-phase
saturation profiles at 0.4 PVI; the bottom-right panel is for wetting-phase saturation profiles at 0.5
PVI.
Next, we compare the RS–NE method with the IMPES method for Case-3 and
Case-4 in terms of time step size and total compute time. In RS–NE, we consider its
two different versions, i.e., the standard and adaptive RS–NE methods. In Case-3,
the tested medium consists of three subdomains with the different permeabilities of 1
md or 100 md, i.e.,


70 6 x 6 100 and 0 6 y 6 100,
1 md,
k = 1 md,
150 6 x 6 180 and 50 6 y 6 150,


100 md otherwise.
The configurations are shown in the top-left panel of Figure 3. The porosity of
this medium φ is 0.2. The relative permeabilities are quadratic function of water
saturation, i.e., β = 2. The capillary pressure parameter Bc is taken as 100 bar.
The injection rate is 0.2 PV/year. Other relevant parameters are provided in Table
1. We run the test on a 100 × 50 mesh using RS with a fixed time step size 4t =
0.02 year. In Figure 3, we show the contour plots of the calculated wetting-phase
saturation at t = 0.2, 0.4, and 0.5 PVI. We can see that the fully implicit approach
successfully resolves the evolution of the wetting-phase saturations, and we find that
the resemblance between the simulated results and the published results in [32] is
remarkable.
In this case, the domain is highly heterogeneous and the capillary pressure functions differ on the different subdomains, leading to the increase in nonlinearity of
the problem that affects the size of the time step. In Table 2, we compare the fully
implicit method with the semi-implicit method in terms of time step sizes and the
computing time. The time step size of IMPES is selected as 4t = 2.5×10−4 such that
it satisfies the requirement of stability. In the adaptive RS–NE method, we set the
initial time step size to be 4t = 0.01 and then adaptively control it by using (3.14).
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B608
HAIJIAN YANG, CHAO YANG, AND SHUYU SUN
Table 2
A comparison of RS–NE and IMPES for Case-3. The mesh is 100 × 50 on four processors.
The simulation is finished at 0.5 PVI. In the adaptive RS–NE method, we set the initial time step
size to be 4t = 0.01.
Number of time steps
Average time step size (years)
Average nonlinear iteration
Execution time (second)
IMPES
104
2.5 × 10−4
−−
137.4
RS–NE
125
0.02
6.4
54.7
Adaptive RS–NE
42
0.061
12.4
30.9
In the standard RS–NE method, the time step size is fixed to 4t = 0.02. We would
like to emphasize that the IPMES method and the standard RS–NE method cannot
obtain smoothed solutions when the time step size becomes larger. It is clear from
Table 2 that the required number of time steps of RS–NE is much smaller than that
of IMPES. Moreover, we can take a larger time step size for the RS–NE method when
an adaptivity mechanism is used to control the time step size, and in this case, the
execution time can be further reduced, although the averaged number of nonlinear
iterations increases. This indicates the new method has better stability and efficiency
than the IMPES method.
In the following, we compare the RS–NE method with the IMPES method for
Case-4. In Case-4, as shown in Figure 4, the domain is also composed of layers of
alternate permeabilities (1 md and 100 md), i.e.,
(
1 md,
30 + 60i ≤ y ≤ 60 + 60i, i = 0, 1, 2, 3,
k=
100 md otherwise.
The wetting phase is uniformly injected across the left-hand side of the layered domain,
which is initially saturated with the nonwetting phase. The production is across
the opposite right-hand side. The injection rate is 0.11 PV/year. Other relevant
parameters are again given in Table 1. In Figure 4, we show the contour plots of
the calculated wetting-phase saturation for the test case at different PVI. As shown
in the figures, the flow behavior with homogeneous capillary pressure is essentially
influenced by the low permeability field and causes a great delay of displacement of
the layer with the smaller permeability value.
We then compare the performance in terms of time step sizes and the computing
time in Table 3. In the adaptive RS–NE method, we again set the initial time step size
to be 4t = 0.01 year and then adaptively control it by using (3.14). In the IMPES
method, the time step size is changed to 4t = 5 × 10−4 year. In the standard RS–NE
method, the time step size is changed to 4t = 0.03 year. From Table 3, it is clear
that the time step size of the fully implicit method is far larger than that of IMPES,
and the total computing time of the fully implicit method is also much smaller than
that of the semi-implicit method. We remark that even with large time step sizes,
the fully implicit method with adaptive time stepping has a similar accuracy as the
semi-implicit method and the fully implicit method with a fixed, small time step size.
From the above tests, we conclude that the RS–NE method, especially the adaptive time-stepping approach, is faster and more robust than the IMPES method in
terms of time step sizes and the computing time. We would like to emphasize that,
with refined spatial resolution, the advantage of the fully implicit method with adaptive time stepping becomes increasingly pronounced, as seen in the weak scaling tests
to be discussed in the last subsection.
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NONLINEAR ELIMINATION FOR TWO-PHASE FLOW
B609
Fig. 4. Case-4: Heterogeneous permeabilities and wetting-phase saturation profiles at different
times. The mesh is 200×100. The top-left panel is for the heterogeneous permeabilities; the top-right
panel is for wetting-phase saturation profiles at 0.2 PVI; the bottom-left panel is for wetting-phase
saturation profiles at 0.4 PVI; the bottom-right panel is for wetting-phase saturation profiles at 0.6
PVI.
Table 3
A comparison of RS–NE and IMPES for Case-4. The mesh is 200 × 100 on four processors.
The simulation is 0.6 PVI. In the adaptive RS–NE method, we set the initial time step size to be
4t = 0.01.
Number of time steps
Average time step size (years)
Average nonlinear iteration
Execution time (second)
IMPES
10909
5 × 10−4
−−
925.1
RS–NE
182
0.03
5.7
630.3
Adaptive RS–NE
75
0.0735
10.4
397.9
Finally, we test the adaptive RS–NE method with a more difficult test case, in
which the permeabilities of the porous medium are random. The configuration of the
test case is listed as Case-5 in Table 1. In the test, the permeabilities of the porous
medium are generated by a geostatistical model using the open source code MRST
(MATLAB Reservoir Simulation Toolbox) [38], ranging from 0.03 to 400 md. The
heterogeneity of the permeabilities often causes unexpected early water breakthrough
and is challenging for the numerical techniques. In the test we set the initial time
step to 4t = 0.02 and run the simulation on a 100 × 100 mesh. The contour plots of
wetting-phase saturation profiles by using the adaptive RS–NE method at different
times are shown in Figure 5. We can see that the proposed approach successfully
resolves the rapid and abrupt evolution of the wetting-phase saturation while keeping
it within the physically meaningful range.
4.2. Comparison of fully implicit methods. In this subsection, we restrict
ourselves to the class of fully implicit methods for Case-3 and Case-4. In the tests,
we first study some parameters that impact the performance of the adaptive RS–NE
method. These parameters include the threshold Nswitch defined in subsection 3.3
for the nonlinear iterations and the adaptivity parameters ρ and η for controlling the
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B610
HAIJIAN YANG, CHAO YANG, AND SHUYU SUN
Fig. 5. Case-5: Heterogeneous permeabilities (log(k), where k has a unit of md) and wettingphase saturation profiles by using the adaptive RS–NE method. The mesh is 100 × 100. The top-left
panel is for the heterogeneous permeabilities; the top-right panel is for wetting-phase saturation
profiles at 0.1 PVI; the bottom-left panel is for wetting-phase saturation profiles at 0.2 PVI; the
bottom-right panel is for wetting-phase saturation profiles at 0.3 PVI.
time step size 4t. We then compare in detail the classical INB and RS methods with
the adaptive RS–NE method.
4.2.1. The choice of parameters in adaptive RS–NE. A threshold Nswitch
is used to determine the moment of freezing the global nonlinear problems and turning
to the nonlinear elimination step, in order to balance the costs and accuracy of solving
the global and subspace nonlinear problems. So in this test we focus on the performance of the adaptive RS–NE method with different thresholds Nswitch for the first
and second time steps, as shown in Table 4. The table presents the total computing
time, the number of global nonlinear iterations, and the number of subspace nonlinear
iterations and, in parentheses, the average number of iterations for solving the global
and subspace Jacobian systems, respectively. The tests are done on a fixed 400 × 200
mesh using eight processors. We set the initial time step size to be 4t = 2.5 × 10−3
for Case-3 and 4t = 5 × 10−3 for Case-4, respectively. It is observed from the table
that (a) when Nswitch is small, such as two, the proposed adaptive RS–NE method
works poorly or even fails to converge; (b) when we increase Nswitch , the number of
the global linear iterations becomes larger, which may degrade the performance of the
RS–NE method. Based on the above observations, we find that the RS–NE method
with the threshold Nswitch around five is appropriate in terms of the total computing
time and the robustness and stability of the solver.
In the adaptive RS–NE method, we adaptively control the time step size 4t by
using the strategy (3.14). In the simulation, ρ ∈ (0, +∞) is a safeguard to avoid
excessive change of the time step size between any two immediate time steps, and
η ∈ (0, 1] is used to control the adjustment of the time step size. Hence, the parameters ρ and η are used to determine the maximum allowable time step size in the
simulation. In Tables 5 and 6, we show the effect of the parameters for Case-3 and
Case-4. It is observed from the results that the average time step size will increase
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NONLINEAR ELIMINATION FOR TWO-PHASE FLOW
Table 4
A comparison of the threshold Nswitch for Case-3 and Case-4. In the table, “Nonlinear”
denotes the number of inexact nonlinear iterations per time step, “Linear” denotes the average
number of GMRES iterations per nonlinear iteration, and “Time” denotes the total computing time
in seconds. “Global” (“Local”) denotes the performance of RS–NE in the global (subspace) iteration
and “1st-step” (“2nd-step”) denotes the performance of RS–NE at the first (second) time step. The
“∗∗” means divergence of the nonlinear iteration.
Case
Nswitch
Case-3
2
5
10
15
2
5
10
15
Case-4
1st-step: Global (Local)
Nonlinear
Linear
Time
∗∗
15 (8)
30.3 (1.0)
29.6
19 (7)
31.2 (1.2)
31.3
26 (6)
34.7 (1.3)
48.6
34 (101)
24.9 (1.0)
134.9
14 (8)
41.5 (1.0)
36.8
19 (6)
42.8 (1.0)
39.8
18 (2)
46.7 (1.0)
42.7
2nd-step: Global (Local)
Nonlinear
Linear
Time
∗∗
15 (12)
26.0 (1.6)
29.1
16 (10)
27.6 (1.7)
29.4
18 (8)
27.7 (1.8)
34.1
18 (34)
22.5 (1.0)
55.8
17 (18)
24.1 (1.0)
36.9
16 (10)
26.4 (1.0)
29.6
17 (8)
28.8 (1.0)
32.5
Table 5
A comparison of ρ and η for Case-3. The initial time step size is 4t = 0.01. 100 × 50 mesh.
η = 0.75
Number of time steps
Average time step size (years)
Average nonlinear iteration
Execution time (second)
ρ = 2.5
Number of time steps
Average time step size (years)
Average nonlinear iteration
Execution time (second)
ρ = 3.5
30
0.0845
17.5
29.8
η = 1.0
31
0.0814
17.5
30.2
ρ = 3.0
35
0.0729
14.8
30.5
η = 0.9
35
0.0726
14.9
29.8
ρ = 2.5
42
0.0610
12.4
30.9
η = 0.7
44
0.0574
11.8
31.2
ρ = 2.0
52
0.0490
10.5
33.7
η = 0.6
50
0.0511
10.7
33.3
ρ = 1.5
68
0.0368
8.3
36.7
η = 0.5
56
0.0453
11.0
38.0
Table 6
A comparison of ρ and η for Case-4. The initial time step size is 4t = 0.01. 200 × 100 mesh.
η = 0.75
Number of time steps
Average time step size (years)
Average nonlinear iteration
Execution time (second)
ρ = 2.5
Number of time steps
Average time step size (years)
Average nonlinear iteration
Execution time (second)
ρ = 3.5
53
0.1031
14.6
433.0
η = 1.0
52
0.1055
15.1
370.9
ρ = 3.0
62
0.0883
12.4
390.4
η = 0.9
60
0.0913
13.3
387.6
ρ = 2.5
75
0.735
10.4
397.8
η = 0.7
80
0.0683
9.3
397.7
ρ = 2.0
93
0.587
9.2
527.9
η = 0.6
93
0.0592
8.9
438.6
ρ = 1.5
125
0.0439
7.9
522.4
η = 0.5
107
0.0512
7.8
465.3
and fewer time steps are required when we use larger ρ and η, in general resulting
in the reduction of the total computing time; meanwhile it leads to the number of
average nonlinear iterations decreases, hence the cost for solving nonlinear problems
may increase. Observed from the tables, we find that ρ ∈ (1.5, 3.5) and η ∈ (0.5, 1]
are appropriate to guarantee the convergence of the proposed method.
4.2.2. A comparison of INB and RS. The price to pay for using implicit
methods for the fully coupled problem is to solve a nonlinear system (3.13) at each
time step. When solving such nonlinear systems using the Newton method or its
many variations, however, we face the numerical challenges arising from physically
feasible saturation fractions between 0 and 1. As shown in Figure 6, the INB method
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HAIJIAN YANG, CHAO YANG, AND SHUYU SUN
4
4
10
2
10
2
Residual
Residual
10
10
−1
−1
10
10
RS
INB
−3
0
RS
INB
−3
10
10
2
4
6
8
10
12
Nonlinear iteration
14
16
18
20
0
2
4
6
8
10
12
Nonlinear iteration
14
16
18
20
Fig. 6. Nonlinear residual history for Case-3 (left) and Case-4 (right) at the first time step
using the INB and RS methods. The time step size is 4t = 0.01.
4
4
10
10
2
2
10
Residual
Residual
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B612
−1
10
10
−1
RS
10
RS−NE−Saturation
−3
−3
10
0
10
2
4
6
RS
RS−NE−Saturation
RS−NE−Pressure
8
10
12
Nonliear iteration
14
16
18
20
0
RS−NE−Pressure
2
4
6
8
10
12
Nonlinear iteration
14
16
18
20
Fig. 7. Nonlinear residual history for Case-3 (left) and Case-4 (right) at the first time step.
In the adaptive RS–NE method, the initial time step size is 4t = 0.01.
fails after eight Newton iterations, which is caused by the floating point exception
from the restriction of the saturation variable. Hence, in the study we use the class
of active-set reduced-space methods to fix this issue. However, along with the simulation moving onward, the nonlinear system becomes higher nonlinearity arising from
heterogeneous permeability of high contrast, strong nonlinearities of relative permeability, and spatially varied capillary pressure functions. As a result, the nonlinear
system is extremely difficult to solve, and the classical RS method is not convergent
when it attempts to do the first nonlinear iteration. The divergence of the nonlinear
iteration for the RS method is caused by the failure of linear search. Hence, we apply
the NE–RS method to resolve the issue.
4.2.3. A comparison of RS and adaptive RS–NE. The RS–NE method
consists of two major ingredients: a subspace correction and a global update. In the
subspace correction step, the effective choice of the components to be eliminated plays
an important role in the success of the whole algorithm.The RS–NE method may be
not convergent when suitable components are not eliminated properly. Hence, we
investigate the influence of the choices on the components to be eliminated. In the
test, we carry out two different strategies: the pressure approach (i.e., removing the
pw -component) and the wetting-phase saturation approach (i.e., removing the Sw component). In Figure 7, we show the results with the choices of removing components
for Case-3 and Case-4 at the first time step. We can see that the pressure approach
cannot make any progress after several global nonlinear iterations are finished and then
we turn to the first subspace correction. This divergence of the nonlinear iteration
for the pressure approach is caused by the failure of linear search. On the other
hand, by using the second strategy, the RS–NE method can make a profit from the
subspace correction, and the number of nonlinear iterations in the global step is
smaller than that for the classical RS method. We would like to point out that the
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B613
NONLINEAR ELIMINATION FOR TWO-PHASE FLOW
3
10
4
2
10
−4
Value
Residual
10
−1
−10
10
10
RS
Gnorm
Lambda
RS−NE
−3
10
−16
0
1
2
3
4
5
6
7
8
Nonlinear iteration
9
10
10
11
5
10
15
20
Line search
25
30
Fig. 8. Case-3: A comparison of RS and adaptive RS–NE. Left panel: Nonlinear residual
history at the second time step. Right panel: History of the step length λ when it attempts to do the
first nonlinear iteration. The mesh is 100 × 50, and the initial time step size is 4t = 0.01.
3
10
4
10
2
10
−4
10
Value
Residual
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10
−1
−10
10
10
RS
−3
10
Gnorm
Lambda
RS−NE
−16
0
2
4
6
Nonlinear iteration
8
10
12
10
5
10
15
20
Line search
25
30
35
Fig. 9. Case-4: A comparison of RS and adaptive RS–NE. Left panel: Nonlinear residual
history at the second time step. Right panel: History of the step length λ when it attempts to do the
first nonlinear iteration. The mesh is 200 × 100, and the initial time step size is 4t = 0.01.
pressure approach always fails to work for two-phase flow problems at other time
steps. Hence, in all the following tests we will choose the second strategy for the
subspace correction.
We next investigate the performance of the adaptive RS–NE method with the
classical RS method at the second time step. In Figures 8 and 9, we show the nonlinear
residual history for Case-3 and Case-4, respectively. From the left panel of the figures,
one can see that the nonlinear system is extremely difficult to solve, and the classical
RS method is not convergent when it attempts to do the first nonlinear iteration. The
divergence of the nonlinear iteration for the RS method is caused by the failure of the
linear search. To figure out the reason for the failure, in the right panel of Figures
8 and 9, we plot the histories of the step length λ (denoted by the blue line) and
the norm of nonlinear residual functions (denoted by the red line) when it attempts
to do the first nonlinear iteration. In the process of the first nonlinear iteration, we
first find the search direction by solving inexactly the Jacobian system, then try to
compute the next approximate solution along this search direction by using a step
length λ. The purpose of step length is to determine how far we should go from the
current approximation and meanwhile make sure the reduction of nonlinear residual
function. However, as seen in the right panel of the figures, the value of the nonlinear
residual function stagnates around 10 without any improvement after dozens of line
searches, even for tiny values of step length with λ = 10−15 , while with the help of
nonlinear elimination, it is clear that RS–NE converges much easier and can reach
convergence with up to 12 iterations for these test cases. So we conclude that the
performance of the RS–NE method is better than that of the classical RS method.
We also remark that the classical RS method begins to fail to converge for any time
steps l > 2.
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B614
HAIJIAN YANG, CHAO YANG, AND SHUYU SUN
6
6
10
10
4
10
2
Residual
Residual
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4
10
10
−1
10
−3
10
0
2
10
−1
10
Mx=100
Mx=200
Mx=400
2
−3
10
4
6
8
Nonlinear iteration
10
12
14
0
Mx=100
Mx=200
Mx=400
2
4
6
8
10
12
14
16
Nonlinear iteration
Fig. 10. Case-4: A comparison of adaptive RS–NE with different mesh sizes. Left panel:
Nonlinear residual history at the first time step. Right panel: Nonlinear residual history at the
second time step.
Finally, we focus on Case-4 and study the convergence behavior when the mesh
size is gradually refined from 100 × 50 to 400 × 200. In the test, the initial time step
size is 4t = 0.02 for the case of a 100 × 50 mesh and then is reduced by half as the
mesh is refined accordingly. Figure 10 shows the nonlinear residual history at the
first and second time steps. It is observed from the figure that as the mesh is refined
the adaptive RS–NE method always converges and the number of nonlinear iterations
only increases very slowly.
4.3. Parallel performance study. Scalability is an important issue in parallel
computing, and the issue is more significant when solving large-scale problems with
many processors. Therefore, we numerically investigate the weak and strong scalabilities of the proposed algorithm for Case-4 in terms of the computing time and
iterations.
A weak scaling test is carried out to examine the performance of the solver when
the problem size is increased in proportion to the number of processor cores. In this
test, we are particularly interested in stopping the calculation at a target simulation
time. In the test, we start with two processor cores and a 128 × 64 mesh, and the
number of processor cores is increased as the mesh is refined accordingly. We show
in Table 7 the results obtained with the fully implicit method with adaptive time
stepping and the semi-implicit method. In the adaptive fully implicit method, the
initial time step size is 4t = 0.02 for the case of a 128 × 64 mesh and then is reduced by half as the mesh is refined accordingly. In the semi-implicit method, the
time step size is fixed to 4t = 10−3 for the case of a 128 × 64 mesh and then is
reduced by half as the mesh is refined accordingly. However, the IMPES method
is not convergent for the cases of a 512 × 256 mesh with 4t = 2.5 × 10−4 and a
1024 × 512 mesh with 4t = 6.25 × 10−5 . Hence, we choose smaller time step sizes for
these two cases. The simulation is terminated at 0.11 PVI. Table 7 gives a comparison between the computing time results of the fully implicit method with adaptive
time stepping and the semi-implicit method. For the semi-implicit method, the total
computing time can increase by more than one order of magnitude, as the mesh is
refined, which is due to the stability restriction on the time step size. For the fully
implicit method, the total numbers of time steps increase as more processor cores
are used. As a result, the total numbers of nonlinear and linear iterations, as well
as the total compute time, increase rapidly as the mesh is refined. It is clear that
the implicit approach still outperforms the semi-implicit approach, especially for finer
spatial resolutions.
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
B615
Table 7
Weak scaling results of the adaptive RS-NE method and the IMPES method. The “−−” means
that the computing time is more than 10 hours.
Number of processors
Mesh size
Initial time step size
Number of time steps (Adaptive)
Average time step size (years)
Average nonlinear iteration
Average linear iteration
Execution time (second)
Number of time steps (IMPES)
Time step size (years)
Average linear iteration
Execution time (second)
2
128 × 64
2 × 10−2
12
0.0918
12.0
2.2
15.7
103
10−3
14.0
20.3
8
256 × 128
10−2
19
0.0553
16.9
14.2
48.0
2 × 103
5 × 10−4
83.4
217.8
32
512 × 256
5 × 10−3
31
0.0331
21.5
22.8
165.2
8 × 103
1.25 × 10−4
192.2
3361.2
128
1024 × 512
2.5 × 10−3
51
0.0196
32.0
31.8
680.8
3.2 × 104
3.125 × 10−5
−−
−−
Table 8
Strong scalability with different number of processors Np for RS–NE.
Mesh
1024 × 512
Np
2
4
16
64
128
32
64
128
256
512
2048 × 1024
Nonlinear
4.4
4.4
4.4
4.4
4.4
6.8
6.8
6.8
6.8
6.8
Linear
4.5
30.5
42.6
46.0
57.8
65.0
65.4
81.2
91.2
111.8
Time
416.6
197.9
63.2
18.8
12.7
388.8
169.8
107.4
72.6
50.1
64
400
32
200
16
100
Speedup
Computing time
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NONLINEAR ELIMINATION FOR TWO-PHASE FLOW
50
8
4
Ideal
Mx=1024
Ideal
Mx=2048
25
12.5
2
4
8
2
16
32
64
Processors
128
256
512
2
4
8
16
32
64
Processors
128
256
512
Fig. 11. Strong scalability results with different number of processors Np for RS–NE.
To further study the parallel scalability of the fully implicit method, we perform
the strong scaling test on two fixed meshes and different numbers of processor cores
in Table 8. The test is run with a fixed time step size 4t = 10−5 and the simulation
is terminated at the fifth time step. We can see from the table that the number of
Newton iterations does not change and the number of linear iterations increases mildly
with growth of the number of processor cores. In Figure 11, we report the computing
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B616
HAIJIAN YANG, CHAO YANG, AND SHUYU SUN
time with respect to the number of processors. When the number of processors Np
increases from 2 to 512, the total computing time decreases at a reasonably good
rate, which indicates that the proposed method has a good speedup for this range of
numbers of processors.
5. Conclusions. In this paper, we have proposed an active-set reduced-space
method with nonlinear elimination for the fully implicit simulation of two-phase flows.
In the proposed algorithm, we present a variational inequality formulation of twophase flow problems to avoid nonphysical undershoot or overshoot of the saturation
fractions, and we employ an active-set reduced-space algorithm to solve the resultant
nonlinear complementarity system arising at each implicit time step. To improve the
robustness of the nonlinear solve, a field-split nonlinear elimination preconditioner is
proposed to remove the high nonlinearity that often triggers the failure of convergence.
Numerical results for several test cases have shown that the proposed algorithm is
robust and superior to the classical IMPES method. Although our discussion in this
paper is restricted to the two-dimensional two-phase flows, we believe that the general
methodology is applicable to the other nonlinear multiphase flow problems in two or
three dimensions.
Acknowledgments. The authors would like to thank the anonymous reviewers
for the valuable suggestions leading to the improvement of the paper.
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